Bonus puzzle studio · geometry constraints and backtracking
Draw Grid Squares from Marked-Vertex Constraints
Turn every marked grid point into a family of possible squares, choose one square for each point, and coordinate the choices so no two square edges occupy the same grid segment.
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Your task: one square for each marked point
Choose one grid-aligned square for every marked point. Each square must have exactly one marked corner, meet that point’s minimum side length, and stay inside the board. Different squares may overlap inside or cross at a point, but may not share a unit edge.
- Start with three squares on a 6×6 board: learn the rules, list candidates, and inspect a worked arrangement in Missions 1–3. Build your own arrangement in Mission 4.
- Extend to nine squares on a 10×10 board: Missions 6–7 show how to start with a point that has few choices and inspect a worked arrangement. Mission 8 is your nine-square construction.
An anchor is the marked point used as a square’s corner. A candidate is one square that meets the rules by itself; it must still fit with the squares chosen for other anchors.
Build candidates first, then select a compatible set
marked point → legal candidate squares → one choice per point → no shared unit edge
All square sides must follow horizontal or vertical grid lines and remain inside the displayed board. One grid gap is one unit; 6×6 means six unit cells across, with seven grid intersections on each boundary. Coordinates start at the top-left: x increases rightward and y downward. A square is not chosen only because it fits. It must use the correct marked vertex, meet the minimum side length, and leave legal choices for every other marked point.
1. AnchorExactly one marked point appears among the square’s four vertices. Other marks may lie inside the square or along an edge away from its corners.
2. SizeWhite anchors need side at least 2; black anchors need side at least 3.
3. EdgesNo two chosen squares may share a positive-length grid-edge segment.
4. Global checkEvery marked point receives one chosen square in this lesson’s explicit original interpretation.
original interpretation made visible
The worked example asks for the same number of squares as marked points, and both printed constructions use every point once. This page therefore makes “one square anchored at each marked point” explicit instead of leaving it implicit.
The worked example forbids overlapping edges, not overlapping interiors. Its worked construction contains interior overlap and isolated edge crossings. Those are allowed here. Sharing a collinear grid segment is not.
black point: side ≥ 3white point: side ≥ 2chosen square edge
Puzzle habit: begin with an anchor that has few candidate squares. A restrictive point often controls the rest of the construction.
1
Decode what is allowed and what is forbidden
Interior overlap is different from edge overlap
Unit-edge test: imagine splitting every grid line into one-unit segments. Two squares conflict only when they claim at least one identical horizontal or vertical unit segment.
2
Generate candidate squares for each original mark
A point may be any one of the four vertices
How do we find every candidate? Use B1, the top black point at (4, 0). Coordinates count units right and down from the top-left of the board.
- Try B1 as each of the four corners. Extending upward leaves the board, so only the two downward directions can work.
- A black anchor needs a side of at least 3. Extending down and right leaves only 2 units before the right boundary, so that direction cannot work.
- Extending down and left leaves 4 units to the left boundary and 6 downward. Only sides 3 and 4 fit. Inspect all four corners of each: B1 is the only marked corner.
| Trial | Keep or reject? |
|---|---|
| Side 2, down and left | Reject: black points need side ≥ 3. |
| Side 3, down and right | Reject: its right edge reaches x = 7, outside the board. |
| Side 3, down and left | Keep: top-left (1, 0), within the board, only B1 marked at a corner. |
| Side 4, down and left | Keep: top-left (0, 0), within the board, only B1 marked at a corner. |
| Side 5 or more, down and left | Reject: the left edge goes beyond x = 0. |
So B1 has exactly two candidates. Use the explorer below to inspect them. Then apply the same corner, size, boundary, and marked-corner checks to W1 and B2.
Candidate explorer
Point type—
Minimum side—
Candidate count—
Viewing—
Choose a point.
| original mark | Coordinate | Type | Minimum side | Legal candidates |
|---|
3
Reconstruct original Figure 28-5
Three side-4 squares satisfy every printed condition
Inspect a complete three-square example. The ledger lists the whole arrangement. Next reveals one square at a time; the diagram counters describe only the squares shown so far. The questions below refer to the complete worked arrangement.
Printed construction
| Step | Anchor | Top-left | Side |
|---|
Squares shown so far0 / 3
Anchors used so far0 / 3
Shared unit edges among shown squares0
Rule: interior overlapallowed
original construction: the top black point, right white point, and bottom black point are each a vertex of a side-4 square. The square interiors overlap, but no pair of boundaries shares a grid segment.
4
Build a legal solution for Worked example 2
Select a mark, then choose one compatible candidate square; a new choice replaces that mark’s earlier square
Your three-square construction. Choose one candidate for each marked point and check that their boundaries share no unit edge. You may return to Mission 3 for help.
Squares placed0 / 3
Anchors covered0 / 3
Used unit edges0
Current statusincomplete
Choose one candidate for each point
Select a marked point.
Place three squares. A blocked candidate shares at least one unit edge with a square already placed.
5
Explain compatibility before claiming uniqueness
Use the local rules to justify a construction; explore all candidates if you want a uniqueness proof.
Reason it through
A square has four vertices, but exactly one must be a given mark. Compare its boundary with each chosen square: no unit edge may be shared. One working arrangement proves existence. Only an exhaustive argument can establish uniqueness.
The worked example prints one answer but does not state that it is unique. This audit generates every legal candidate, then checks every one-per-point combination for shared edges.
Not run yet.
Legal candidates—
Raw triples—
Compatible complete triples—
Worked solution unique?—
Nine candidate squares
| ID | Anchor | Top-left | Side |
|---|
Backtracking: choose one candidate for the most restricted anchor, reject every candidate sharing an edge, and continue only along compatible branches.
6
Read original Practice 2 and find its restrictive anchors
A 10×10 grid carries three black points and six white points
Now solve a larger puzzle. This board has nine anchors, so you need nine squares. Generate candidates with the same method. For example, a side-2 square extending right and down from W1 at (4, 0) must be rejected: W2 at (6, 0) would be a second marked corner. Start your search with an anchor that has few candidates.
Practice candidate explorer
Black points3
White points6
Total candidates56
Rarest anchorB2 · 5
Selected type—
Minimum side—
Candidate count—
Viewing—
Choose a point.
| Point | Coordinate | Type | Minimum side | Candidates |
|---|
Rarest-first strategy: the black point on the right boundary has only five candidates—the fewest on this board. Testing that point early reduces branching.
7
Reconstruct the worked example’s nine-square reference arrangement
The later answer section supplies one compatible placement
Inspect one complete nine-square solution. The ledger describes all nine squares even before you reveal them. The counters describe only the squares shown so far; the questions refer to the complete worked arrangement.
Reference placement ledger
| Step | Anchor | Top-left | Side |
|---|
Squares shown so far0 / 9
Largest side shown so far—
Sum of side lengths shown so far0
Shared unit edges among shown squares0
Reference answer: the nine sides are 3, 2, 4, 3, 5, 3, 2, 2, and 3 in the reveal order. Their sum is 27, so the nine square perimeters use 108 distinct unit-edge segments.
8
Solve Practice 2 and compare every complete arrangement
Any compatible nine-square construction is accepted
Your nine-square construction. Choose one compatible square for every anchor. Any complete arrangement satisfying the rules is accepted. The complete-solution gallery is an optional comparison after your attempt.
Squares placed0 / 9
Anchors covered0 / 9
Used unit edges0
Statusincomplete
Practice square builder
Select a marked point.
Choose one candidate per anchor. The builder prevents shared unit edges.
Added exact-search result. Under the explicit one-square-per-point interpretation, the board has 56 legal candidates and exactly four complete compatible arrangements.
Candidates—
Raw one-per-anchor combinations—
Complete arrangements—
Viewing solution—
Complete-arrangement gallery
Run the audit to compare the four arrangements.
9
Marked-vertex square workshop
Eight objective checks · all eight correct complete the mission
10
Exit ticket and certificate
Five correct responses complete Lesson 28.2